Let \(R_p(I)\)  be the rotation poset of \(I\) . There is a 1-1 correspondence between closed subsets of \(R_p(I)\)  and the stable matchings of \(I\)  [1]}.
Let \(S\)  be a maximum profile closed subset of the rotation poset \(R_p(I)\) , whose rotation profiles are reversed and negated, and let \(M\)  be the unique corresponding stable matching. Then by addition over reversed negated profiles described in the text above this lemma, \(p_r^{\prime }(M) = p_r^{\prime }(M_0) + \sum _{\rho _i \in S} p_r^{\prime }(\rho _i)\) .
Suppose \(M\)  is not generous. Then there is a stable matching \(M^{\prime }\)  such that \(p_r^{\prime }(M^{\prime }) \succ p_r^{\prime }(M)\) . \(M^{\prime }\)  corresponds to a unique closed subset \(S^{\prime }\)  of the rotation poset, such that \(p_r^{\prime }(M^{\prime }) = p_r^{\prime }(M_0) + \sum _{\rho _i \in S^{\prime }} p_r^{\prime }(\rho _i)\) . But \(p_r^{\prime }(M^{\prime }) \succ p_r^{\prime }(M)\)  and so \(S\)  cannot be a maximum profile closed subset of \(R_p(I)\) , a contradiction.
